cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

A121680 a(n) = [x^n] (1 + x*(1+x)^(n+1) )^(n+1).

Original entry on oeis.org

1, 2, 12, 76, 655, 6816, 81690, 1109816, 16782399, 278438740, 5016899833, 97368894756, 2021749249403, 44658312247290, 1044437050070340, 25757381769393392, 667470006331599523, 18119105978249333988
Offset: 0

Views

Author

Paul D. Hanna, Aug 15 2006

Keywords

Comments

a(n) is divisible by (n+1): a(n)/(n+1) = A121681(n).

Examples

			At n=4, a(4) = [x^4] (1 + x*(1+x)^5 )^5 = 655, since
(1 + x*(1+x)^5 )^5 = 1 + 5*x + 35*x^2 + 160*x^3 + 655*x^4 +...

Crossrefs

Cf. A121681; variants: A121673-A121679.

Programs

  • Mathematica
    Table[Sum[Binomial[n+1,k] * Binomial[(n+1)*k,n-k], {k,0,n+1}], {n,0,20}] (* Vaclav Kotesovec, Oct 02 2020 *)
  • PARI
    a(n)=sum(k=0,n+1,binomial(n+1,k)*binomial((n+1)*k,n-k))

Formula

a(n) = Sum_{k=0..n+1} C(n+1,k) * C((n+1)*k,n-k).

A121674 a(n) = [x^n] (1 + x*(1+x)^n )^n.

Original entry on oeis.org

1, 1, 5, 28, 233, 2376, 28102, 379016, 5707025, 94439440, 1699067321, 32951077193, 684009742319, 15110032165151, 353485501643471, 8721374385748256, 226128389777924385, 6142306518887606112, 174311816444805024379
Offset: 0

Views

Author

Paul D. Hanna, Aug 15 2006

Keywords

Examples

			At n=4, a(4) = [x^4] (1 + x*(1+x)^4 )^4 = 233, since
(1 + x*(1+x)^4 )^4 = 1 + 4*x + 22*x^2 + 76*x^3 + 233*x^4 +...

Links

Crossrefs

Cf. variants: A121673, A121675-A121680.

Programs

  • Mathematica
    Table[Sum[Binomial[n,k] * Binomial[n*k,n-k], {k,0,n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 12 2015 *)
  • PARI
    a(n)=sum(k=0,n,binomial(n,k)*binomial(n*k,n-k))

Formula

a(n) = Sum_{k=0..n} C(n,k) * C(n*k,n-k).

A121675 a(n) = [x^n] (1 + x*(1+x)^(n+1) )^n.

Original entry on oeis.org

1, 1, 7, 43, 371, 3926, 47622, 654151, 9999523, 167557174, 3046387103, 59616689595, 1247357472869, 27747682830531, 653192297754076, 16206706672425167, 422358302959175123, 11526119161103900834
Offset: 0

Views

Author

Paul D. Hanna, Aug 15 2006

Keywords

Examples

			At n=4, a(4) = [x^4] (1 + x*(1+x)^5 )^4 = 371, since
(1 + x*(1+x)^5 )^4 = 1 + 4*x + 26*x^2 + 104*x^3 + 371*x^4 +...

Crossrefs

Cf. variants: A121673, A121674, A121676-A121680.

Programs

  • Mathematica
    Table[Sum[Binomial[n,k] * Binomial[(n+1)*k,n-k], {k,0,n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 12 2015 *)
  • PARI
    a(n)=sum(k=0,n,binomial(n,k)*binomial((n+1)*k,n-k))

Formula

a(n) = Sum_{k=0..n} C(n,k) * C((n+1)*k,n-k).

A121676 a(n) = [x^n] (1 + x*(1+x)^(n-1) )^(n+1).

Original entry on oeis.org

1, 2, 6, 32, 250, 2412, 27524, 360600, 5296050, 85805420, 1515794467, 28926900312, 591903009295, 12907255696636, 298428274844730, 7284351640977920, 187013495992710210, 5033669346061547724, 141643700005223732471
Offset: 0

Views

Author

Paul D. Hanna, Aug 15 2006

Keywords

Comments

a(n) is divisible by (n+1): a(n)/(n+1) = A121677(n).

Examples

			At n=4, a(4) = [x^4] (1 + x*(1+x)^3 )^5 = 250, since
(1 + x*(1+x)^3 )^5 = 1 + 5*x + 25*x^2 + 85*x^3 + 250*x^4 +...

Crossrefs

Cf. A121677; variants: A121673-A121675, A121678-A121680.

Programs

  • PARI
    a(n)=sum(k=0,n+1,binomial(n+1,k)*binomial((n-1)*k,n-k))

Formula

a(n) = Sum_{k=0..n+1} C(n+1,k) * C((n-1)*k,n-k).

A121678 a(n) = [x^n] (1 + x*(1+x)^n )^(n+1).

Original entry on oeis.org

1, 2, 9, 52, 425, 4236, 49294, 655096, 9731313, 159114880, 2832245911, 54400757016, 1119436524947, 24532373640334, 569732648555295, 13962373137304496, 359767723241891425, 9715902692094061488
Offset: 0

Views

Author

Paul D. Hanna, Aug 15 2006

Keywords

Comments

a(n) is divisible by (n+1): a(n)/(n+1) = A121679(n).

Examples

			At n=5, a(5) = [x^5] (1 + x*(1+x)^5)^6 = 4236, since
(1+x*(1+x)^5)^6 = 1 + 6*x + 45*x^2 + 230*x^3 + 1050*x^4 + 4236*x^5 +...

Crossrefs

Cf. A121679; variants: A121673-A121676, A121680.

Programs

  • PARI
    a(n)=sum(k=0,n+1,binomial(n+1,k)*binomial(n*k,n-k))

Formula

a(n) = Sum_{k=0..n+1} C(n+1,k) * C(n*k,n-k).

A121679 a(n) = A121678(n)/(n+1) = [x^n] (1 + x*(1+x)^n )^(n+1) / (n+1).

Original entry on oeis.org

1, 1, 3, 13, 85, 706, 7042, 81887, 1081257, 15911488, 257476901, 4533396418, 86110501919, 1752312402881, 37982176570353, 872648321081531, 21162807249523025, 539772371783003416, 14433746294326451095
Offset: 0

Views

Author

Paul D. Hanna, Aug 15 2006

Keywords

Examples

			At n=5, a(5) = [x^5] (1 + x*(1+x)^5)^6/6 = 4236/6 = 706, since
(1+x*(1+x)^5)^6 = 1 + 6*x + 45*x^2 + 230*x^3 + 1050*x^4 + 4236*x^5 +...

Crossrefs

Cf. A121678; variants: A121673-A121676, A121680.

Programs

  • Mathematica
    Table[Sum[Binomial[n+1,k] * Binomial[n*k,n-k] / (n+1), {k,0,n+1}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 12 2015 *)
  • PARI
    a(n)=sum(k=0,n+1,binomial(n+1,k)*binomial(n*k,n-k))/(n+1)

Formula

a(n) = Sum_{k=0..n+1} C(n+1,k) * C(n*k,n-k) / (n+1).

A382859 a(n) = Sum_{k=0..n} binomial(n,k) * binomial((n-1)*(k+1),n-k).

Original entry on oeis.org

1, 1, 5, 37, 345, 3851, 49468, 713931, 11391985, 198523495, 3741919446, 75702725440, 1633591960883, 37404262517506, 904734768056239, 23030071358784701, 614912094171482849, 17172036245893988575, 500281954849350450946, 15170753984617328108901
Offset: 0

Views

Author

Seiichi Manyama, Apr 07 2025

Keywords

Links

Crossrefs

Main diagonal of A381425.
Cf. A121673, A121674, A121675.

Programs

  • Magma
    [&+[Binomial(n,k) * Binomial((n-1)*(k+1),n-k): k in [0..n]]: n in [0..21]]; // Vincenzo Librandi, Apr 09 2025
  • Mathematica
    Table[Sum[Binomial[n,k] * Binomial[(n-1)*(k+1),n-k], {k,0,n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 07 2025 *)
  • PARI
    a(n) = sum(k=0, n, binomial(n, k)*binomial((n-1)*(k+1), n-k));

Formula

a(n) = [x^n] (1 + x/(1-x)^n)^n.

A121677 a(n) = A121676(n)/(n+1) = [x^n] (1 + x*(1+x)^(n-1) )^(n+1) / (n+1).

Original entry on oeis.org

1, 1, 2, 8, 50, 402, 3932, 45075, 588450, 8580542, 137799497, 2410575026, 45531000715, 921946835474, 19895218322982, 455271977561120, 11000793881924130, 279648297003419318, 7454931579222301709
Offset: 0

Views

Author

Paul D. Hanna, Aug 15 2006

Keywords

Examples

			At n=4, a(4) = [x^4] (1 + x*(1+x)^3 )^5/5 = 250/5 = 50, since
(1 + x*(1+x)^3 )^5 = 1 + 5*x + 25*x^2 + 85*x^3 + 250*x^4 +...

Crossrefs

Cf. A121676; variants: A121673-A121675, A121678-A121680.

Programs

  • Mathematica
    Flatten[{1,Table[Sum[Binomial[n+1,k] * Binomial[(n-1)*k,n-k] / (n+1), {k,0,n+1}], {n, 1, 20}]}] (* Vaclav Kotesovec, Jun 12 2015 *)
  • PARI
    a(n)=sum(k=0,n+1,binomial(n+1,k)*binomial((n-1)*k,n-k))/(n+1)

Formula

a(n) = Sum_{k=0..n+1} C(n+1,k) * C((n-1)*k,n-k) / (n+1).

A121681 a(n) = A121680(n)/(n+1) = [x^n] (1 + x*(1+x)^(n+1) )^(n+1) / (n+1).

Original entry on oeis.org

1, 1, 4, 19, 131, 1136, 11670, 138727, 1864711, 27843874, 456081803, 8114074563, 155519173031, 3189879446235, 69629136671356, 1609836360587087, 39262941548917619, 1006616998791629666, 27044968746461571213
Offset: 0

Views

Author

Paul D. Hanna, Aug 15 2006

Keywords

Examples

			At n=4, a(4) = [x^4] (1 + x*(1+x)^5 )^5 /5 = 655/5 = 131, since
(1 + x*(1+x)^5 )^5 = 1 + 5*x + 35*x^2 + 160*x^3 + 655*x^4 +...

Crossrefs

Cf. A121680; variants: A121673-A121679.

Programs

  • Mathematica
    Table[Sum[Binomial[n+1,k] * Binomial[(n+1)*k,n-k] / (n+1), {k,0,n+1}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 12 2015 *)
  • PARI
    a(n)=sum(k=0,n+1,binomial(n+1,k)*binomial((n+1)*k,n-k))/(n+1)

Formula

a(n) = Sum_{k=0..n+1} C(n+1,k) * C((n+1)*k,n-k) / (n+1).
Showing 1-9 of 9 results.

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